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Abstract

Emergency service providers are facing the following problem: how
and where to locate vehicles in order to cover potential future demand
effectively. Ambulances are supposed to be located at designated locations such
that in case of an emergency the patients can be reached in a time-efficient
manner. A patient is said to be covered by a vehicle if (s)he can be reached by
an ambulance within a predefined time limit. Due to variations in speed and the
resulting travel times it is not sufficient to solve the static ambulance
location problem once using fixed average travel times, as the coverage areas
themselves change throughout the day. Hence we developed a multi-period version,
taking into account time-varying coverage areas, where we allow vehicles to be
repositioned in order to maintain a certain coverage standard throughout the
planning horizon. We have formulated a mixed integer program for the problem at
hand, which tries to optimize coverage at various points in time simultaneously.
The problem is solved metaheuristically using variable neighborhood search. We
show that it is essential to consider time-dependent variations in travel times
and coverage respectively. When ignoring them the resulting objective will be
overestimated by more than 24%. By taking into account these variations
explicitly the solution on average can be improved by more than
10%.

1. Introduction

1.1. Motivation

Much research has been focused on solving various kinds of vehicle
location problems in both static and dynamic environments. Travel times
however, which should form the basis for quantifying the objective to be
optimized, mostly are assumed to be constant throughout the planning
horizon. Variations in velocity and speed during peak hours usually are
completely ignored. Recent developments in Global Positioning System (GPS)
allow velocities – so called Floating Car Data (FCD) – to be
collected over time. Based on past observations time-varying velocities can
be defined for different street segments within a street network. As a
consequence time-dependent travel times can be estimated for different time
periods. Taking into account time-dependent travel times significantly
influences the quality of the solution obtained. Especially in areas with
high traffic volume during peak hours travel speed and the resulting travel
time can vary significantly throughout the day. Just using (averaged) fixed
travel times is no longer sufficient and will lead to sub-optimal or
infeasible solutions. In order to overcome this issues we allow vehicles to
be relocated throughout the planning horizon in an anticipatory manner in
order to tackle variations in speed and the resulting changes in
coverage.

Please note that the proposed model at hand can be applied to other
applications (e.g. locating a fleet of taxis, service technicians,
break-down service vehicles, etc.), where quality is usually measured in
terms of coverage or response times, as well. It can be used to manage a set
of service entities (i.e. ambulances, service vehicles, taxis), that need to
be located such that customer entities (i.e. patients, customers,
passengers, etc.) can be reached efficiently in case of a request.

1.2. Related work

1.2.1. Early models

Most ambulance location models found in the literature are
extensions of the location set covering model (see Toregas et al., 1971), which try to
minimize the number of ambulances necessary in order to cover all demand
points. Another early model for the problem is the so called maximal
covering location problem and was originally proposed by Church and ReVelle (1974), which tries
to maximize the total demand covered given a fleet of fixed size. These
are static models that do not consider the fact that ambulances will
become unavailable throughout the day and certain demand points might
not be covered any more. Several approaches have been developed in order
to handle this. One possibility includes multiple coverage, i.e. demand
points are supposed to be covered by more than one vehicle. Such a
model, called the double standard model (DSM) was introduced by
Gendreau et al. (1997).
Other possibilities for handling vehicles becoming unavailable include
busy fractions, where the probability that vehicles might become
unavailable is modeled explicitly (see Daskin
(1983)).

1.2.2. Static double standard model

The DSM maximizes the demand covered by at least two vehicles,
implicitly taking into account the fact that vehicles might become
unavailable, while ensuring certain additional requirements concerning
coverage are met. In particular they try to ensure that
α% of the population are covered within
r1 time units and all demand
points need to be covered by at least one vehicle within
r2 time units, where
r1 < r2.
The model has been proposed in Gendreau et
al. (1997) and has been solved using tabu search.
Their formulation considered a single period only and treated travel
times as static. It did not allow the resulting changes in coverage
throughout the day to be taken into account. Another drawback of the
single-period DSM formulation is that it does not limit the demand that
can be covered by one single vehicle. Especially in densely populated
areas defining coverage in terms of a radius within which patients shall
be reached might not be sufficient. The DSM has been further extended by
Doerner et al. (2005). In
their paper they deviate from the traditional assumptions that vehicles
can cover any emergency that lies within a predefined radius by taking
into account capacity considerations. At most a certain amount of demand
(i.e. number of inhabitants in the designated areas) can
reasonably be covered by one single
vehicle.

Successful applications of models based on DSM formulations include
one by Thirion (2006) to two
provinces in Wallonia, Belgium. The author has shown that a significant
improvement in coverage could be reached without increasing the number
of ambulances in use. Doerner et al.
(2005) applied an extended version of the DSM to data
coming from the eight rural provinces in Austria. A broader overview on
different applications can be found in Laporte et al. (2009). Furthermore models have been
applied successfully to data from Montreal, Canada (Gendreau et al., 1997). Their model has
also been extended into a dynamic environment (Gendreau et al., 2001). They take
advantage of the available time between consecutive calls by
anticipating future decisions on the redeployment of the
fleet.

1.2.3. Dynamic coverage models

Several approaches already have been proposed that handle the
ambulance location problem in a dynamic setting. An extension of the
maximum expected coverage location problem proposed by Daskin (1983) has been developed by
Repede and J Bernardo
(1994). The authors propose and solve a multi-period
maximum expected coverage location problems with time-variant travel
times and changing fleet sizes (TIMEXCLP). Their objective is to
optimize the expected coverage at various points in time. They
incorporate temporal variations in the daily demand process in addition
to spatial variations and multiple states of vehicle availability.
However, the resulting number of relocations of vehicles in between is
neglected and not considered explicitly during the stage of
optimization. In their paper Repede and Bernardo propose a decision
support system. First the location of ambulances will be determined
using the TIMEXCLP model, which then will be tested by means of a
simulation model. If required system characteristics are not met a new
TIMEXCLP will be generated. Another multi-period model for dynamic
demand environments which minimized the number of ambulances required
while meeting predetermined ambulance availability requirements has been
proposed in K Rajagopalan et al.
(2008). In contrast to previous models, ambulance
specific busy probabilities have also been considered. They solved the
model using tabu search. The solution then is validated by means of
simulation. Only demand nodes that are covered with a certain
probability are included in the system wide expected coverage. The model
itself is solved iteratively starting from the first time period for a
given fleet size. If coverage requirements are (not) exceeded the model
will be rerun using a decremented (incremented) fleet size. The model
itself however is solved iteratively for consecutive time periods, while
ignoring the resulting relocations from the ambulance’s point of
view. In Gendreau et al.
(2006) a probabilistic model has been proposed, which
also limits the number of reallocations allowed. A broad overview on
different location problems and their applications in the context of
ambulance location problems can be found in Brotcorne et al. (2003) and Laporte et al.
(2009).

Empirical evidence against time-dependent variations in traveling
times encountered by fire engines is given in Kolesar et al. (1975), who validated a
model for average fire engine travel times, depending on the distance
traveled for the city of New York. They conclude that the time of day
only has minor impact on average response velocities. A similar analysis
has been repeated and extended for ambulances in Budge et al. (2010) for the city of
Calgary. Despite this empirical evidence, the operating experience of
practitioners in Austria indicates that travel speeds do vary over the
course of a day, for the following reasons:

• Streets in Europe, as opposed to streets on the North
American continent, typically tend to be narrower.
Especially during peak times the traffic volumes in urban
areas increases and causes congestion. Due to narrower
streets ambulances, as regular vehicles, are affected and
cannot circumnavigate jammed vehicles in congested street
segments.

• Next one needs to point out that not every request
ambulances have to serve entitles them to have right of way
by all means. Only in case of life-threatening incidences
and calls where the response time has medical consequences
ambulances are allowed to make use of acoustic and light
signals and – as a consequence – to speed up
their drive.

• Even if ambulances are allowed to make use of their
acoustic and/or light signals they are still bound to legal
regulations concerning the maximum travel speed, which must
not be exceeded. As the resulting accident rate would
increase disproportionately, drivers furthermore are
required not to drive unreasonably, endangering themselves
or other road-users.

Due to these observations we were explicitly encouraged to
investigate the ambulance location and relocation problem with
time-varying travel times, as observed by our industry partner. The
corresponding variations are significant and should be taken into
account for a proper deployment of ambulances.

1.3. Contribution

The DSM is an ideal formulation for solving the tactical problem under
consideration. Stochastic demand of vehicles that can serve at most one call
at a time can be handled by backup coverage (see Hogan and ReVelle, 1986). By enforcing double coverage
of patients the dynamics of the underlying systems are already partially
considered. Hence it is highly suitable for optimizing the location of
ambulances in real-world. In order to overcome some of its drawbacks we have
decided to extend their model in several ways.

(i) We have extended the single-period model to a multi-period
version. Solving a single-period DSM once only taking into
account current or average travel times is not
satisfactory.

(ii) We will explicitly take into account time-dependent
variations in speeds and the resulting changes with respect to
the corresponding coverage.

(iii) Furthermore we will allow vehicles to be relocated in order
to overcome this issue, rather then finding a static position
for the entire planning horizon. Relocating too many vehicles
should be avoided and hence this issue will be considered in the
objective function.

(iv) Finally, we extend the model of Gendreau et al. (1997) relating to capacity
considerations by further developing what was proposed by
Doerner et al.
(2005). In Doerner
et al. (2005), however,
all demand points that are located
within a certain distance are considered for this kind of
capacity consideration. In this paper, however, we explicitly
assign fractions of patients’ demand to vehicles in order
to model this feature more correctly. By using this approach,
potential patients being counted more than once for capacity
considerations can be avoided.

By ignoring time-dependent variations in travel times one severely
misestimates the resulting coverage, which will lead to inferior solutions.
By taking into account those variations during the optimization phase and
allowing vehicles to be relocated, the quality of the solutions obtained can
be increased significantly. The goal is to provide the decision maker with
time-dependent location plans, such that the resulting coverage can be kept
at the required level throughout the planning horizon. These plans are
especially useful when dispatchers do not have access to an online decision
support system. First, we take advantage of the availability of
time-dependent data in order to get a clearer view of the traffic situation
and the resulting changes in coverage throughout the day. Next we introduce
a model which incorporates this information and allows vehicles to be
repositioned to optimize the coverage at several points in time. Finally we
developed an innovative metaheuristic search procedure, based on the
concepts of VNS.

There are different ways to consider time-dependent data. First we
solve our model in a myopic way at various points in time using the
prevailing travel times respectively. All resulting relocations will be
calculated ex post. Rather then solving the model
independently several times, we try to solve the model simultaneously for
various points in time. All resulting relocations will be considered
implicitly during the optimization phase.

2. Problem formulation

Let W be the set of potential vehicle locations. The set of demand
locations is denoted by V. The problem itself is defined on a graph G=(V∪W,E), where E is the set of edges {(j,i):j∈W,i∈V}, each associated with travel time sjit prevailing at time t. Each demand location
i∈V is associated with a corresponding demand value
di. The goal
is to locate p vehicles among all potential vehicle
locations, such that all demand locations can be reached within
r2 time units and
α % of the population (0 < α < 1) can be reached within
r1 (r1 < r2) time units at all
times t∈T, where T={1,…,T}. No more than
pj
vehicles can be located at location j∈W at any time. The objective is to maximize the demand covered by at
least two vehicles within a radius of r1
at all t∈T. The number of vehicles to be located at vehicle location j(j∈W) in t∈T is denoted by yjt. The binary decision variable xik,t will be equal to one if the demand at vertex i∈V is covered k times
(k = 1, 2) at time t. Let Wi1,t(Wi2,t) be the set of vehicle locations from which patient
i can be reached within radius
r1
(r2) at time
t. (Wik,t={j∈W:sjit⩽rk;k=1,2}). The set of patients that can be reached within
r2 time units by a vehicle located
at location j∈W is denoted by Vj2,t. In order to define the set of patients to be covered by locations,
the shortest path was considered and then evaluated in terms of the prevailing
speed at various points in time. The subset of time intervals excluding the very
last one is denoted by T′. The number of vehicles that are supposed to be relocated from
location i to j(i,j∈W) between t and t + 1 (tT′) is denoted by
rijt. In order to consider the end-of-horizon-effects, the relocation of
vehicles in period t = T influences the location of vehicles in period
t = 1. The number
of vehicles to be relocated between T and 1 is denoted by
rijT.

In order to emphasize locating more vehicles in densely populated areas we
assume that any given vehicle can cover at most ω
inhabitants reasonably. The actual demand at patient
location i∈V that is covered by a vehicle located at j∈W in t is denoted by zijt.

The DSM introduced by Gendreau et al.
(1997) furthermore has been extended in terms of
time-dependent travel times. We consider them as well as the resulting
variations in coverage levels and all relocations necessary. The resulting model
is a multi-period DSM (mDSM). Rather than optimizing the total demand covered at
least twice within r1, the coverage
throughout the entire planning horizon and all necessary relocations will be
considered simultaneously. In the objective function (see Eq. (1)) we sum up the two objectives using a
weighted sum approach. The penalty for relocating vehicles is denoted by
β.

maxf(x)=∑t∈T∑i∈Vdixi2,t-β∑i,j∈Wrijt,

(1)

s.t.∑j∈Wi2,tyjt⩾1∀i∈V,t∈T,

(2)

∑i∈Vdixi1,t⩾α∑i∈Vdi∀t∈T,

(3)

∑j∈Wi1,tyjt⩾xi1,t+xi2,t∀i∈V,t∈T,

(4)

xi2,t⩾xi1,t∀i∈V,t∈T,

(5)

∑j∈Wyjt=p∀t∈T,

(6)

yjt⩽pj∀j∈W,t∈T,

(7)

yjt+∑i∈Wrijt-∑i∈Wrjit=yjt+1∀j∈W,t∈T′,

(8)

yjT+∑i∈WrijT-∑i∈WrjiT=yj1∀j∈W,

(9)

∑i∈Vj2,tzijt⩽ωpj∀j∈W,t∈T,

(10)

∑j∈Wi2,tzijt=di∀i∈V,t∈T,

(11)

xik,t∈{0,1}∀i∈V,t∈T,k∈{1,2},

(12)

yjtinteger∀j∈W,t∈T.

(13)

Throughout the planning horizon T a DSM has to be considered simultaneously at every instance
t∈T, while considering the prevailing travel times. Constraints
(2) ensure that every demand
location i will be covered at least once within
r2 at every point in time
t. Constraints (3) ensure that α% of the total
demand is covered within r1 at every
instance of time t. The combination of Constraints
(4) and (5) ensure that a
demand location can only be covered once (twice) if sufficient vehicles are
located around demand location i∈V. A total number of p vehicles has to be
located (see Constraints (6)), while
ensuring that capacity restrictions at the individual potential waiting sites
are not exceeded (see Constraints (7)).
Constraints (8) and (9) ensure
that resulting relocations of vehicles between different location can take place
accordingly. Due to Constraints (10) and
(11) the demand of every patient will be assigned to
locations, while making sure that no single vehicle can reasonably cover more
than ω patients. The variables xik,t modeling the coverage are supposed to be binary (see Constraints
(12)). The number of vehicles to
be located is integral (see Constraints (13)).

Due to the changing traffic situation a feasible solution can not be
guaranteed. In order to solve the problem Constraints (2), (3) and (11) will be relaxed. The
following extended problem formulation will be used, including Constraints
(4)–(9),
(10), (13), where
x+ = max{0,x}.

maxF(x)=f(x)-∑t∈T(γ1f1t(x)+γ2f2t(x)+γ3f3t(x)),

(14)

where

f1t(x)=∑i∈V1-∑j∈Wi2,tyjt+∀t∈T,

(15)

f2t(x)=α∑i∈Vdi-∑i∈Vdixi1,t+∀t∈T,

(16)

f3t(x)=∑i∈V(di-∑j∈Wi2,tzij)+∀t∈T.

(17)

By reducing the number of time intervals T
to one, the problem at hand reduces to a static myopic problem, where the
situation at a particular point in time, considering prevailing travel times
only and without any relocations, will be optimized.

3. Variable neighborhood search

The problem at hand will be solved by methods inspired by means of VNS. A
basic sketch of VNS is depicted in Algorithm 1. The algorithm stops after a
certain number of iterations or as soon as a given time limit is reached. During
the shaking phase, various neighborhoods are used to explore the solution space
thoroughly and in order to escape local optima. Departing from the best solution
x a neighboring solution
x′ will be generated at random from neighborhood
Nk(x).
The embedded local search phase tries to improve any solution
x′ obtained after the previous shaking step. If
the resulting solution x″ is not worse than the
best solution x, it will be accepted and becomes the new
best solution x and the shaking phase will be restarted
with the first neighborhood, otherwise the search continues with neighborhood
k + 1. Due to the
fact that many location solutions lead to the very same objective function
value, the best solution will change more frequently. Therefore the concept of
accepting deteriorating solutions is not considered as important or
useful.

3.1. Variable neighborhood search for double standard
model

Any solution for a DSM can be uniquely represented by the location of
vehicles and the assignment of patients to locations respectively. The
search procedure will be initialized by assigning vehicles to locations
randomly. The assignment of patients to locations will be determined using a
simple greedy procedure.

3.1.1. Initial solution

To start with all vehicles will be located randomly across all
potential locations, while making sure capacity restrictions at those
locations are obeyed.

3.1.2. Shaking

During the shaking phase the solution space will be explored more
thoroughly. Two different shaking operators with resulting six
neighborhoods
Nk
(k = 1 … 6) have been
implemented and tested. The first shaking operator
(move), responsible for neighborhood
structure
Nk
(k = 1 … 3), tries to
move vehicles currently located at
k + 2
locations. Therefore a subset of locations S (where |S|=k+2) will be determined randomly. Vehicles will then be moved
among the locations j within the subset S, while making sure the capacity restrictions at the involved
locations j∈S are obeyed.

3.1.3. Local search

After the solution x has been perturbed
using the corresponding neighborhood structure, the resulting solution
x′ will tried to be further improved by
means of local search. Within the local search phase any solution is
tried to be further improved by means of small local changes.
We’ve tested two local search operators. The main idea of our
first local search operator is to move a single
vehicle from one location to a different one. Using a first-improvement
strategy any vehicle currently located at location
j is tried to be inserted at all other
locations j′
(j′ ≠ j). Our second local search
operator tries to swap as many vehicles as
possible between two locations i and
j. Both operators are executed in a
first-improvement fashion. Only feasible movements – in terms of
the capacity restrictions prevailing at the potential waiting sites
– are considered.

Any solution for our mDSM can be uniquely represented by the location
of the vehicles and the assignment of patients to vehicle locations. The
position of all vehicles at time t is referred to as
a location pattern. A location pattern for every time
period t is used in order to represent a solution for
the entire planning horizon. For runtime reasons the assignment of patients
to vehicle locations will be determined using a simple greedy
procedure.

3.2.1. Initial solution

In order to initialize the VNS for the mDSM the problem will be
solved in a myopic fashion and independently, once for every time period
t∈T. A DSM (using the procedures described in Section
3.1) will be solved at
each step, taking into account the current travel times and
corresponding coverage radii.

3.2.2. Shaking

Any two location pattern can be compared in terms of the Hamming
distance (Hamming, 1950),
where the number of necessary relocations to get from one location
pattern to the other one is given by one half of the Hamming distance.
In order to explore the solution space thoroughly a shaking operator
with three resulting neighborhood structures has been designed. At first
two consecutive time periods t1
and t2 (where
t2 = t1 + 1) are randomly selected and evaluated in
terms of the resulting coverage and the necessary number of relocations.
In every shaking step one location pattern will be
assimilated to another one. Any assimilation
tries to make two location patterns more similar to each other in terms
of the number of vehicles placed at all possible locations, hence
reducing the Hamming distance. The decision which location patterns
should be assimilated to each other is based on the following
considerations.

First the solution quality of the location patterns
lp(t1) (lp(t2))
prevailing at t1
(t2) will be evaluated and
compared to the solution of the myopic problem
lpM(t1)
(lpM(t2)).
The time period in which the current location pattern performs worse
(best) with respect to the corresponding myopic location pattern will be
denoted by t−
(t+). Furthermore the number
of relocations involved in order to move from location pattern
lp(t1)
to lp(t2)
will be determined by evaluating in terms of the Hamming distance.
Depending on the solution quality and the number of relocations involved
the following strategy will be chosen: In case the number of relocations
necessary between the two location patterns prevailing at
t1 and
t2 is
high (as compared to the average number of
relocations to be executed per day) an attempt will be made to overcome
this issue. The location pattern which currently performs worse
lp(t−)
in terms of the resulting coverage will be assimilated to the
other location pattern
lp(t+)
respectively. Hence emphasis will be put on improving the solution in
terms of the necessary number of relocations while taking into account a
potential decrease in the resulting coverage. In the case of a
small Hamming distance between the two
location patterns
lp(t1)
and
lp(t2),
the pattern which currently performs worse
(lp(t−))
relative to its corresponding myopic solution will be assimilated to the
location pattern of the myopic solution
(lpM(t−))
prevailing at time t−.
Hence more emphasis will be put on the improvement of the solution in
terms of the resulting coverage.

In every shaking step location pattern
lp1 will be assimilated to
lp2. Any neighborhood
Nk
(k = 1 … 3) changes the
number of vehicles placed according to location pattern
lp1 at up to
10·k% of all possible locations. The
new number of vehicles to be placed at location j
will be randomly drawn from the interval minvjlp1,vjlp2,maxvjlp1,vjlp2, where vjlp refers to the number of vehicles currently located at
location l according to location pattern
lp. As the total number of vehicles in the
resulting pattern lp1∗ might have changed, location pattern lp1∗ might need to be revised such that exactly
p vehicles are located at all locations in
total. Hence excess (missing) vehicles are removed from (inserted into)
the location pattern afterwards.

3.2.3. Local search

Within the local search phase any solution is tried to be further
improved by means of small local changes. The main idea of our local
search operator, similar to what has been done for the DSM (see Section
3.1.3), consists of moving
single (or swapping several) vehicles from one location to a different
one. In order to emphasize a diversification strategy at the very start
of the solution process, the local search will limit itself to the
location patterns currently prevailing at
t1 and
t2. Hence allowing a large
number of neighborhoods to be explored without the burden of extensive
local search phases in between. During the solution process however the
search process will be intensified by steadily extending the number of
time intervals considered within local search.

4. Computational experiments

4.1. Data description

For our computational experiments we used real-world data from the city
of Vienna (Austria). The population of Vienna amounts to approximately 1.7
million inhabitants. The locations of potential patients were derived from
census data. The 3920 demand points i∈V were derived by aggregating the population by squares (250
meter × 250 meter), each with
a demand (measured in terms of inhabitants) between 1 and 2977. We are going
to solve the problem inspired by an ambulance service provider in Vienna.
The goal is to provide adequate coverage for the population in case of
emergency. A real-world road network from Teleatlas using all streets
accessible by car has been used. FCD was used in order to estimate
time-dependent variations in travel time on each link over the course of
time.

The FCD-data were aggregated from data collected by a fleet of taxis in
the city of Vienna, communicating their speed and positions throughout
several months by means of GPS. The raw data has been aggregated by Austrian
Institute of Technology which provided us with the average travel speed on
each link within 96 intervals, each with a length of 15 minutes, starting at
midnight. The average travel speed along the individual links varied by up
to 25% from the overall average throughout the course of the day. Especially
in the core of the city the average speed decreased significantly during
rush hours. A graphical representation of the changes concerning the traffic
situation throughout the day is given in Section 4.2.5. We use this data in order to solve the problem at
hand at a tactical level. Only observations collected during weekdays were
used.

Please note that the data were collected by a fleet of taxis, which
serves as an estimate for the travel times experienced by ambulances. Unlike
regular cars they are allowed to use designated lanes. This however is
perfectly suitable for our application at hand, as ambulances are also
allowed to use those special lanes within the network. The total planning
horizon of 24 hours was equally split into 6 time intervals and
time-dependent travel times were aggregated accordingly. Analyzing the data
led to the conclusion that a more fine-grained level of discretization is
not necessary (see Section 4.2.2).

The algorithm was tested on various scenarios, each with a different
number of potential locations spread throughout the city of Vienna. Testcase
1 considered the 16 potential waiting sites currently in use. For testcases
2, 3 and 4 the number of potential waiting sites was changed to 51, 94 and
163 respectively. In testcase 2 the shops of a large fast food chain are
used as potential vehicle location sites. Testcase 3 comprises all police
stations in Vienna as potential vehicle locations and in testcase 4 a
selection of typical Austrian luncheonettes is used. There are no legal
regulations concerning coverage standards in Austria. The model uses the
rules set by the United States Emergency Medical Services Act of 1973, where
r1 (α) are set to 10 minutes (95%). The
second coverage radius, r2, was set
to 20 minutes. A total number of 14 vehicles had to be located at various
potential waiting sites, each having a capacity pj=1(j∈W). For a topographical representation of the four testcases under
consideration see Fig.
1(a)–(d).

4.2. Results

4.2.1. Importance of time-dependent travel
time

In order to emphasize the importance of time-dependent travel times
and the resulting coverage radii the following experiment was set up
first: The DSM formulation (using the approach described in Section
3.1) was solved once for
the entire planning horizon using average travel times only. By
completely ignoring time-dependent variations in travel times, this
naive approach optimizes the location of
vehicles using average travel times only. The resulting coverage
throughout the day was evaluated ex post, by
explicitly taking into account the prevailing travel times given the
naive location pattern obtained before. On the other hand we solved the
DSM several times in a myopic fashion taking into
account the current travel times respectively. A designated run time
limit of 10 seconds per time interval was used in order to solve the DSM
using VNS.

The results obtained represent average values over five independent
runs per testcase. Fig.
2 shows the average
solutions (dotted line) obtained by the VNS for the DSM if solved in a
naive way (i.e. when assuming fixed average travel times). The DSM has
been solved once for the entire planning horizon using
average travel times only. Afterwards we
evaluated the resulting solutions using the current travel times in
order to access the solutions’ robustness to time-dependent
variations in travel times and coverage radii respectively (dashed
line). The obtained solutions are compatible during off-peak hours but
fail to perform well during peak hours. This clearly shows the
importance of time-dependent data. Neglecting time-dependent variations
will lead to serious over-estimation of the resulting coverage. In fact
during peak hours the resulting objective is overestimated by
24.35%.

Furthermore we show the average solutions (solid line) obtained for
the mDSM when using the myopic approach and
optimizing each point in time independently. The deterioration during
peak hours cannot be avoided, but the situation can be improved
significantly. By explicitly taking into account time-dependent
variations in travel times and speed the solutions obtained using the
naive approach can be improved by up to 10.2% .

The VNS in use is capable of obtaining (near-) optimal solutions.
See Section 4.2.3 for a
detailed comparison of the solutions obtained by VNS as compared to
CPLEX.

4.2.2. Effects of level of aggregation

In the most detailed level of aggregation a potential patient
corresponds to the set of inhabitants living in a square of length 250
meter. In order to decrease the size of the underlying model we tried to
aggregate demand nodes into larger super-nodes. Depending on the level
of aggregation a a super-node was obtained by
combining squares of a2 original
demand nodes. At first sight the solutions obtained using aggregated
patients nodes seem to be insensitive to changes in the underlying level
of aggregation. However when re-evaluating the solution obtained in
terms of the underlying original patient nodes (at the finest level of
detail available), it can be shown that on average the solution is
overestimated dramatically. This effect is commonly referred to as the
optimality error (see Francis et al.
(2009) for a detailed analysis of errors associated
with aggregation in the context of location models). Fig. 3
shows the resulting misestimation of the obtained solutions when
changing the level of aggregation. The level of aggregation
a is plotted on the horizontal axis. The
resulting number of aggregated demand nodes in shown in parenthesis.
When a is set to 2 on average the resulting
solution is overestimated by 18.5% (i.e. the corresponding solution for
the aggregated problem suggests an objective function value that is
18.35% above the solution, when the corresponding optimized location
pattern is evaluated in terms of the original demand nodes at the finest
level of detail). When further increasing the level of aggregation to 5
the resulting solution will be overestimated by 26.7%. In order not to
misestimate the resulting quality of the solution and get an adequate
level of realism it is essential to use the data at the finest level of
detail.

Overestimation of solution quality with varying level of aggregation
a. (The resulting number of aggregated demand nodes
is given in parenthesis.)

4.2.3. Solving the double standard model

In order to demonstrate the effectiveness of the proposed VNS
algorithm for the DSM (as described in Section 3.1) the problem was first solved using
CPLEX. The total run time limit for the execution of the model was set
to 10 hours. All relevant results can be found in Table 1.
The first two columns indicate the test instance (n) as well as the
number of potential waiting locations (|W|). The solution
(Fmax),
the resulting coverage
(C(Fmax)),
the best bound
(bmin)
as well as the time necessary for finding it
(tmax)
are shown in the first part of the table. The problem then was solved
using VNS. The run time limit was set to 10 seconds. The best and
average solutions found are indicated in the two columns labeled
Fmax
and Favg
respectively. The best solutions found are italicised. The column headed
by
C(Favg)
denotes the average coverage, measured in terms of the percentage of all
inhabitants that are covered by at least two vehicles. The necessary run
times (in seconds) until the best solution was found are given by
tavg.
Because of the random nature of the proposed algorithm five independent
test runs, using a different stream of random numbers, were executed per
instance. As a benchmark value an upper bound on the possible coverage,
assuming that an ambulance is located at every location (i.e.
p = W), is given in the column headed
by C¯.

Please note that the value of the objective function
(F) is made up by several components. The
original objective is to optimize coverage. Violating constraints types
2, 3 and 10
would lead to a penalization in the objective function value as
indicated by Eqs. (15)–(17). The DSM focuses on one single time
period only. Hence no relocations and penalties therefore will
occur.

The results show that the implemented VNS is a robust algorithm
finding high quality solutions in a reasonable amount of run time. Given
the short run time limit of 10 seconds per instance for the VNS the
algorithm was able to find the same solution as CPLEX for the smaller
instances 1 and 2. For the remaining two instances the best solutions
found by VNS within a couple of seconds outperformed the ones found by
CPLEX in ten hours. Please note that the resulting coverage is almost
equivalent to the value of the objective function obtained, as the
penalties associated with the three aforementioned soft constraints are
very small.

4.2.4. Solving the multi-period double standard
model

One of the drawbacks of the single-period DSM formulations is that
the resulting number of relocations is not explicitly considered and
most likely will result in a high number of relocations. Obviously the
location problem at hand cannot be solved independently at various
points in time. Hence the mDSM, considering the single-period
DSM-approach for several points in time plus all resulting relocations
between time periods, has been solved using both CPLEX and VNS (as
described in Section 3.2).
Table 2 shows the results obtained. In order to emphasize the
effectiveness of the proposed algorithm an experiment was set up in
which the entire time horizon corresponding to one single day (24 hours,
starting from midnight) was equally split into six time periods, each of
length four hours. The total run time designated for the execution of
VNS was set to 60 seconds. The penalty β for
relocating vehicles was set to 10000. Hence relocating a vehicle would
be considered beneficial if an additional number of 10,000 (equivalent
to 0.58 % of the total population) inhabitants would be covered twice
within r1. Please note that
β is independent on the time necessary
to relocate the vehicle or the distance involved. For a more detailed
evaluation of the choice of β see
Table 4. The penalties
γ1,
γ2 and
γ3 for the violation
of the relaxed constraints were set to 2, 0.5 and 0.25 respectively. The
problem at hand was also solved using CPLEX, whose execution was aborted
after ten hours. We show the best bound
(bmin),
best solution
(Fmax),
the consequential coverage
(C(Fmax))
as well as the resulting number of relocations
(rmax)
obtained by CPLEX. The results obtained by means of VNS are shown in the
last five columns. The best and average solution obtained can be found
in the column labeled
Fmax
and Favg
respectively. The average coverage is denoted by
C(Favg).
The average run times (in seconds) until solutions have been found using
VNS is denoted by
tavg.
Again the VNS was executed five times per instance and the results have
been averaged accordingly. The average resulting number of relocations
per day is denoted by
ravg.
The best solutions obtained are italicised.

The mDSM cannot be solved by means of CPLEX in a reasonable amount
of run time. For easier test instances 1 and 2 (where the number of
potential locations is set to 16 and 51 respectively) the solution
obtained by CPLEX within 10 hours is slightly (less than 0.26%) better
then the best solution obtained by VNS within one minute. For the larger
instances however (with 94 and 163 potential waiting sites) CPLEX fails
to find meaningful solutions. The negative values can be explained by
the severe violation of Constraints (2, 3 and 10, which leads to a high penalization in
the objective function value due to Eqs. (15)–(17). The maximum
gap between the best and average solution found by VNS is as small as
0.03% (0.6%) for the smaller (larger) instances. The maximum gap between
the best bound found by CPLEX within 10 hours and the best solution
found by our VNS is as small as 3.4%. This clearly indicates that the
proposed algorithm is highly robust and works extremely well for the
problem under consideration. One can clearly see that the coverage
obtained deviates significantly from the value of the objective
function. This is due to the fact that the imposed coverage standards
can not be satisfied, when considering the actual time-dependent travel
times and the resulting changes in coverage throughout the day. When
considering only average travel times (as done in Section 4.2.3) it seems misleadingly that those
violations would not be an issue. Hence one thinks oneself safe ignoring
time-dependent variations and the imposed coverage standards can not be
met using the true traffic situation.

4.2.5. Spatial patterns of solution

The capital of Vienna is located in the north-east of Austria. The
shape of the city can be approximated by a circle. The majority of the
population lives close to the center of the city, whereas the outskirts
of the city are less densely populated. Most of the inhabitants (83.6%)
live on the south-western side of the river Danube. On average the speed
along streets close to the center of the city is lower than for the rest
of the city. Fig.
4(a) and (b) depict
the traffic situation (in terms of the prevailing speeds) during peak
and off-peak hours. Areas with lower average travel times are shaded in
light gray. These figures also shows a typical solution to testcase 4,
where the location of vehicles are indicated by dark squares. During
peak hours the center of the city tends to be heavily congested,
resulting in an even larger decrease of travel times downtown. In order
to guarantee an acceptable level of coverage during peak hours vehicles
should be relocated towards the city center, however during off-peak
hours the traffic situation downtown recovers and vehicles tend not to
be concentrated on the center of the city only, but are also located in
more remote areas.

4.2.6. Myopic vs. multi-period double standard model
solution

We show that the quality of the solutions obtained can be improved
significantly by explicitly taking into account time-dependent
variations in travel time and the resulting loss of coverage. When
locating vehicles in an anticipatory manner, the resulting number of
relocations can be improved, while keeping the quality of the solution
from deteriorating too much. Table
3 compares the
solutions obtained when solving the location problem in a myopic
fashion, compared to the solution of the mDSM. The planning horizon of
one day (24 hours) was equally split into 6 time intervals of length 4
hours each. All solutions have been obtained by means of VNS. The
results shown are averaged over all 4 test instances, each having been
executed five times independently. The myopic solution obtained is shown
in column
Favg.
Note that for solving the model in a myopic fashion the static location
problem has been solved independently for each time interval, taking
into account the prevailing travel times respectively. The total run
time limit was set to 10 seconds per time interval. Column
ravg
shows the average number of resulting relocations, which had been
evaluated ex post by solving a simple
transportation problem. When solving the mDSM however the problem was
only solved once, while explicitly taking into account the resulting
number of relocations during the stage of optimization. The penalty
value β for relocations was set to 10000.
The resulting average values of the objective function per time interval
with (without) taking into account the cost for relocating vehicles is
shown in column Favg(FavgnoR). The average total number of relocations per day can be
reduced from 20.9 to 10.5, which corresponds to a decrease of 50.0%. The
quality of the solution (ignoring the costs for relocations) only
deteriorates marginally by 0.36%.

4.2.7. Effects of the choice of the penalty value for relocations
(β)

In order to take into account time-dependent variations and the
resulting relocations of vehicles explicitly the following experiment
was conducted. A total run time limit of 10 seconds was set for the
execution of VNS. The resulting average solution obtained, which was
averaged over five independent runs per test case, is shown in
Fig. 5. The mDSM (using the approach described in Section
3.2) was solved given
different values for β. The penalty value
for relocating vehicles was varied between 0 and 150,000 respectively.
For generating the initial solutions for the mDSM a total run time limit
of 0.5 seconds was set. We have plotted the average solution
(Favg)
with respect to the prevailing penalty value
β. Obviously when the costs for
relocating vehicles is set to 0, the solution approaches the one
obtained when solving the mDSM in a myopic fashion. The higher
relocations of vehicles will be penalized, the less frequent these
relocations will happen, which will lead to a deterioration of the
solution.

Average solution
(Favg)
with respect to varying cost per relocation
(β).

If the penalty value β for relocating
vehicles is set to zero, the solution obtained approaches the one
obtained if the model is solved in a myopic fashion, i.e. independently
at designated times, without taking into account resulting relocations
during the stage of optimization explicitly. The higher relocations will
be penalized, the lower the amount of relocations that are going to take
place. Similarly also the quality of solution decreases. Please note
that β indicates the additional number of
inhabitants that would need to be covered twice within
r1 in order to justify an
additional relocation. By reducing β from
100,000 to 25,000 the solution improves from 8.85 × 106 to 9.07 × 106.

A more detailed overview on the effects of parameter
β is shown in Table 4. The average
solution quality throughout the day
(Favg),
as well as the resulting coverage
(C(Favg))
and average number of reallocations
(ravg)
are shown. The total run time limit was set to ten seconds. The shown
values are averages which have been obtained from five independent test
runs.

5. Conclusions and outlook

The availability of FCD allowed us to take advantage of time-dependent
data, analyze their benefits, develop new models and design innovative solution
concepts for the new problem at hand. Furthermore we showed that ignoring
time-dependent information can lead to misleading conclusions. Especially when
it comes to meeting coverage standards one tends to think oneself
safe.

We have extended the single-period DSM for the allocation of a fleet of
ambulances to a set of potential waiting locations, such that the resulting
coverage can be maximized. In the first step the single-period formulation has
been extended in terms of capacity considerations for vehicles. When solving the
DSM one tries to optimize coverage, which is usually measured in terms of
potential patients that can be reached within a predefined time limit. The time
necessary to reach patients in case of an emergency depends on the prevailing
speed of the vehicles in the network, which in turn is highly variable
throughout the day. Hence solving the DSM only once taking into account current
or average travel times is not satisfactory. Therefore we developed a
formulation for the mDSM, a multi-period version of the DSM, which explicitly
takes into account time-dependent variations of travel times and the resulting
changes with respect to coverage. We allow vehicles to be repositioned in order
to respond to changing coverage circumstances throughout the day in an
anticipatory manner. With the mDSM we manage to optimize the tradeoff between
the coverage at different point in times and the resulting number of
relocations.

A metaheuristic algorithm based on VNS has been developed and tested using
real-world data from the city of Vienna. The total population has been clustered
in terms of 4000 demand locations. The number of potential waiting sites of the
ambulances was varied between 16 and 163 in order to place a fleet of 14
vehicles. The proposed algorithm using VNS for solving the DSM works extremely
well, given the combinatorial complexity of the problem at hand. On average it
took only 5 seconds to find a solution which was at least as good as the ones
found by CPLEX within 10 hours. The mDSM cannot be solved exactly using CPLEX in
a reasonable amount of run time. Good solutions have been found using VNS. But
other local search based (meta-) heuristics may work as well. The proposed VNS
for the mDSM allows both a diversification and intensification strategy.
Location patterns of consecutive time periods that are already good in terms of
the resulting number of relocations are tried to be improved in terms of the
resulting coverage and vice versa.

Especially in areas with high variations in travel speed and times
throughout the day it is essential to consider these variations explicitly. This
kind of data (FCD) is readily available and its highly important to take
advantage of this knowledge in order to build real-life models, as the coverage
will be interpreted misleadingly otherwise. By taking into account these
variations the quality of the solution can be improved by up to 10%. In order to
achieve this improvement vehicles should not be located at the same locations
throughout the entire day. Rather they should be repositioned in order to handle
time-dependent variations in coverage. We have shown that relocating vehicles
reactively can help to improve the coverage standards during the day. When
solving the problem in a myopic fashion on average 21.1 vehicles need to be
relocated (see Table 4, where
β = 0). The
quality of the solution – in terms of the resulting average coverage
however – on average only decreases slightly if the resulting average
number of relocations is cut in half (see Table
4, where β = 10,000). By allowing only 10 relocations per day in a
mDSM framework, the quality of the solution on average only deviated by 0.2%
from the myopic solution.

So far this approach is based on the static location problem, taking into
account time-dependent variations of travel times throughout the course of the
day. In the future we would like to also consider this problem in a dynamic
setting, where emergency calls occur and vehicles need to be dispatched in order
to cope with their demand respectively. So far we have used the number of
inhabitants as a proxy for the demand in terms of potential patients. In the
future we are planning to incorporate temporal and spatial variations of the
requests to be generated.

Acknowledgements

Financial support from the FWF
(Translational Research) under Grant L510-N13 is gratefully acknowledged. Special thanks go to the
Austrian Institute of Technology for the preparation of FCD-data. For the street
network of Vienna the data provided by Teleatlas was used. We would also like to
thank two anonymous referees for their valuable comments that considerably helped us
to improve the quality of this paper.

References

Bräysy O. A reactive variable neighborhood search for the
vehicle routing problem with time windows. INFORMS Journal on Computing. 2002;15(4):347–368.